1. Introduction
Let be a stochastically continuous stationary maxstable process with Fréchet marginals . Here maxstable means that the finite dimensional distributions (fidi’s) of are maxstable multivariate distributions, see e.g., [3, 4, 5]. In the following is a nonnegative process satisfying and has spectral process , i.e., we have the de Haan representation, see e.g., [3, 6] (below means equality in law)
(1.1) 
where is a Poisson point process (PPP) on with intensity independent of ’s which are independent copies of . Since we consider here only stationary maxstable processes, adapting the terminology of [4] (motivated by [7]) we shall call the spectral process a BrownResnick stationary process.
The assumption that is stochastically continuous implies that it has a separable and measurable version, see e.g., [8]; the same holds for , see [6]. Therefore in the following we suppose that both and are jointly measurable and separable. In the sequel we shall assume further that has locally bounded sample paths, and thus by (1.1) also has locally bounded sample paths. According to [6], this assumption is important for conditions that guarantee the existence of a dissipative Rosiński (or also called a mixed moving maxima) representation of .
By separability and the locally boundedness of the sample paths of and both and
are welldefined and finite random variables for any
. Further, by (1.1) given we have (see e.g., [9, 10])(1.2) 
Since by measurability of , for any using Fubini theorem
with the Lebesque measure on , we have further that
and
(1.3) 
The above shows that does not depend on the particular choice of the spectral process but only on . By the stationarity of it follows that
for any . Hence is subadditive and thus by Fekete lemma
(1.4) 
Moreover, from the above we conculde that does not depend on the particular choice of the spectral tail process but only on the stationary maxstable process . Referring to [11], is the socalled generalised Pickands constant defined with respect to a BrownResnick stationary process . In [12] is introduced for the lognormal process
(1.5) 
where
is a centered Gaussian process with stationary increments, continuous sample paths and variance function
. Taking to be a fractional Brownian motion (fBm) with selfsimilarity Hurst index , we get that is the classical Pickands constant, see e.g., [13, 14, 15, 11]. The only known values of are 1 and corresponding to and , respectively. In the case of Lévy processes the Pickands constant appears explicitly in [16, 17, 18, 11]. Moreover, for the discretetime case we have that (introduced similarly as for the continuoustime, see [19]) is the extremal index of the stationary time series . In this context, it has been also studied in [20] using the spectral representation of. A huge amount of research is dedicated to calculation and estimation of the extremal index of regularly varying time series, see e.g.,
[21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and the references therein.The main question that arises for Pickands constants , or in the discrete setup for the extremal index of a stationary maxstable time series is:
Q1: Under what conditions are these constants positive or equal to 0?
For the maxstable this question is partially answered in [11] for such that almost surely, and for being a nonnegative random variable in [31]. Specifically, the positivity of has been shown under the assumption that
(1.6) 
In view of [6] since has locally bounded sample paths, then under (1.6) has a dissipative Rosiński representation which is equivalent with being generated by a nonsingular dissipative flow, see [32, 33, 34, 6] for more details. As shown in [11, 31], if has càdlàg sample paths and (1.6) holds, then
(1.7) 
with equality shown under some technical assumptions for both
Gaussian and Lévy case. The investigation therein was motivated by [35, 9].
The former contribution showed that
(1.7) holds with equality for an fBm.
Since in (1.4) is defined as a limit, it turns out that the explicit calculation of is for general too difficult. However, if (1.7) holds with equality, then being an expectation,
can be efficiently simulated. Indeed this has been successfully implemented for the classical Pickands constants in [35].
An interesting question that arises here is:
Q2: Does (1.7) hold with equality for general BrownResnick stationary ?
Clearly, if , then (1.4
) means the convergence in probability
(1.8) 
whereas when we have the convergence in distribution
(1.9) 
For being a symmetric stable () stationary process with the above convergence has been shown in the seminal articles [1, 2], see the recent contributions [36, 37, 38, 39, 40, 41, 42] for related results and new developments.
The findings of [1, 2] are important for the maxstable processes too,
which is already pointed out in [20] for discrete maxstable processes.
Indeed, using the link between maxstable and processes established
in [32] and [43] independently, it follows that
when is generated by a nonsingular conservative flow, which by [6] (under the assumption of locally boundedness of sample paths of ) is equivalent with then we have
(1.10) 
Note that (1.10) holds also when we consider the discrete case , which can be shown for instance by utilising the expression of Pickands constant (which in this case coincides with the extremal index, [19, 31]) derived in [20].
We conclude that is positive if and only if
. Moreover, as shown in [20] for the discrete case,
can be explicitly given in terms of the spectral functions.
Hence question Q1 has a simple answer, namely if is stationary maxstable process with locally bounded sample paths,
then, by [6],
if and only
(1.11) 
Clearly, the convergence in probability in (1.8) implies that
as for any
and a similar implication holds for when (1.9) is satisfied. For being an random field the recent contribution [44] strengthened those convergences to that of for , i.e., showing the uniform integrability of whenever . The case that is a stationary maxstable random field is easier to deal with, see Proposition 3.13 in Section 4.
Our main interest in this contribution is the derivation of expressions for
in terms of the spectral process that appears in the de Haan
representation (1.1). In particular, motivated by Q2, we
show that (1.7) (or a modification of it) holds with equality under
(1.11) without further assumptions. Recall that so far it is only known that the inequality in (1.7) holds for having a dissipative Rosiński representation.
As already shown in [11, 31], different representations for relate to different dissipative Rosiński representations of . Therefore, our analysis is also concerned with such representations for .
Our study of Pickands constants (together with the criteria for its positivity) allows us to investigate the growth of the expectations of and as . The latter can be investigated under the further assumption of the BrownResnick model, i.e., when is a lognormal process. Moreover, for the BrownResnick model an extension of the celebrated Slepian inequality is possible, see Theorem 3.1 below.
Organisation of the paper: Our main results are displayed in Section 2 followed by discussions and some extensions displayed in Section 3. Proofs are postponed to Section 4; an Appendix concludes this contribution.
2. Main Results
Let be as in the Introduction being jointly measurable, separable and with locally bounded sample paths. By the measurability of we have that is a random variable in , see [8]. Write next instead of for an event with . Fixing we have the following splitting formula
(2.1) 
If (1.11) holds, then by [45][Lem 16] the random process defined by
is BrownResnick stationary.
Since has also locally bounded sample paths and almost surely, the corresponding maxstable process is generated by a nonsingular conservative flow. Moreover,
is also a BrownResnick stationary process which is generated by a
nonsingular dissipative flow, provided that (1.11) holds. In order to omit technical details, we refer the reader to the deep contributions [34, 6] for details on conservative and dissipative parts of maxstable processes.
Consequently, by the discussions in the Introduction, condition (1.11) implies that
(2.2) 
Therefore, in the following we can reduce our analysis by considering only the case that , i.e., is generated by a nonsingular dissipative flow. In view of [6] this is equivalent with having a dissipative Rosiński representation i.e., for some nonnegative random process (called also random shape function) which is continuous in probability (we can consider here therefore to be jointly measurable and separable) and for some we have
(2.3) 
where is a PPP on with intensity , independent of ’s which are independent copies of .
In view of [46][Thm 4.2], equation (2.3) implies that for any random variable with density
(2.4) 
is a valid spectral process for , where is independent of and . See below Theorem 3.11 for a converse result. We state next the main result.
Theorem 2.1.
Let be a stochastically continuous maxstable process with de Haan representation (1.1). If has locally bounded sample paths with some spectral process satisfying condition (1.6), then there exists some jointly measurable and separable nonnegative random shape function such that (2.4) holds and moreover we have
(2.5) 
Remark 2.2.
i) If is a stationary maxstable process, then the Pickands constant is defined by (see e.g., [11])
In view of [6] has a dissipative Rosiński representation if and only if . Under this condition, as in the proof of Theorem 2.1, it follows that the Pickands constant is given by
(2.6) 
ii) When has a dissipative Rosiński representation, it is possible to construct such that almost surely, see [6]. Hence by (2.5) for such random shape functions we have
We shall show below that it is also possible to construct such that almost surely, and thus by (2.5) we obtain an alternative formula, namely
For simplicity we shall assume in the following that both and have càdlàg sample paths. Let be the space of càdlàg functions equipped with a metric which makes it complete and separable, see e.g., [47] for details. Let be the Borel algebra on defined by this metric and let be a probability measure given by
Since is a Polish metric space (complete and separable), we can determine a stochastic process with càdlàg sample paths and probability law ; refer to as the spectral tail process. This terminology is in agreement with [22] since as shown in [31], defined above agrees with the spectral tail process of the stationary time series , see [48, 49] for earlier and further results.
By [31][Thm 4.1] all the fidi’s of the maxstable process are determined by . Namely, we have the following infargmax formula valid for ’s positive constants and ’s in (see also Lemma 5.1 in Appendix for a short derivation)
(2.7) 
Consequently, defines and viceversa, from we can calculate the fidi’s of by the generalised Pareto distributions of , see [31][Remark 6.4] and [50] for representations of generalised Pareto distributions.
The next result gives an explicit construction for the random shape function and confirms (1.7).
Theorem 2.3.
Example 1. Consider the Gaussian case with as in (1.5), where is a centered Gaussian process with stationary increments, continuous sample paths and variance function .
We can assume without loss of generality (see [4]) that . Hence almost surely and for the corresponding spectral tail process we simply have . Since for this case , then under (1.11)
In view of [4], the following condition
(2.10) 
implies (1.1) and thus has a dissipative Rosiński representation with random shape function . Moreover
(2.11) 
which has been proved in [11][Thm 2] under additional (redundant) assumptions on .
Example 2. Stationary maxstable Lévy–Brown–Resnick processes are introduced in [17], where the spectral process is constructed from two independent Lévy processes. Specifically, let be a Lévy process with Laplace exponent being finite for . Write for another independent Lévy process with Laplace exponent
Then we set , and if .
In view of [17] the maxstable process with unit Fréchet marginals corresponding to the spectral process is stationary.
Furthermore, by [18] the LévyBrown–Resnick process admits a dissipative Rosiński representation and thus Theorem 2.3 and [11][Thm 3.2] imply that (note that since almost surely then )
(2.12) 
where is the bivariate Laplace exponent of the descending
ladder process corresponding to .
If is a spectrally negative Lévy process, by [17]
,
which agrees with the fact that
3. Discussions & Extensions
3.1. Slepian inequality for BrownResnick maxstable processes
Slepian inequality is essential in the theory of extremes and sample path properties of Gaussian and related processes. Besides it is also useful in numerous fields of mathematics including optimisation and number theory problems, see e.g., [51].
A commonly used version of Slepian inequality given for instance in [52][Thm 1.1] is as follows: If are two centered Gaussian processes, then for any
we have
provided that for all
(3.1) 
Moreover, in view of [53][Eq. 6] (applied to ) for any realvalued function
(3.2) 
Let be maxstable processes with spectral processes . By (1.2) if are separable with locally bounded sample paths such that (3.1) holds, using (3.2) we obtain
(3.3) 
Maxstable processes that are constructed from lognormal Gaussian spectral processes are commonly referred to in the literature as BrownResnick maxstable processes, a first example is given in [7], see [4, 54, 5, 55, 56] for more details and interesting statistical models.
Both processes and are BrownResnick stationary, see [4, 32, 57, 58, 10] if defined by
where are two centered Gaussian processes with stationary increments and variance functions and , respectively. Since in general is different from we cannot use the refinement of Vitale [53] to Slepian inequality stated in (3.2) to arrive at (3.3).
Our next result states the Slepian inequality for BrownResnick stationary processes and . Moreover, it implies a comparison criteria for the corresponding Pickands constants.
It is wellknown that the law of ’s depends only on their variograms , . Therefore we can suppose without loss of generality that for .
Theorem 3.1.
Let be two stationary maxstable BrownResnick processes with spectral processes and , respectively. Suppose that and are separable with locally bounded sample paths. If further for any
(3.4) 
then for any compact set
and moreover
Remark 3.2.
i) As noted in [43], it is not known if there exists
any centered Gaussian process with stationary increments and variance function such that
.
ii) Under the setup of Theorem 3.1, if
, then by Example 1, and consequently . This implies that cannot be generated by a nonsingular conservative flow, therefore we can conclude that
iii) Theorem 3.1 extends the findings of [12][Thm 3.2].
iv) Slepian inequality has been extended to nonGaussian setup in [52].
Using the results of the aforementioned paper, Slepian inequality for maxstable processes can also be derived when and have fidi’s satisfying the conditions of [52][Thm 3.1].
3.2. Wills functional of Gaussian processes and asymptotic constants
In this section, motivated by Vitale [59], we discuss Wills functional for Gaussian processes
and its relation with Pickands and Piterbarg constants.
Consider as in the previous section, where is a centered
separable Gaussian process with variance function and bounded sample paths. In [59] the Wills functional
is defined as
(3.5) 
This functional is important for derivation of lower bounds on supremum of Gaussian processes and random fields. Specifically, by [59][Thm 1] for any we have
(3.6) 
where the last inequality follows by the boundedness of sample paths, see e.g., [60][(3.1)]. Our findings on Pickands constants, when is BrownResnick stationary can be utilised to derive asymptotic properties of Wills functionals as well as lower bounds for . Indeed, the Pickands constant is given in terms of Wills functional as
(3.7) 
As shown in [58] is BrownResnick stationary if and only if has stationary increments. In view of Example 1, we have:
Lemma 3.3.
If is a centered separable Gaussian process with stationary increments, locally bounded sample paths and variance function , then
(3.8) 
provided that (2.10) holds.
Remark 3.4.
Sharp bounds on the expectation of maximum of Gaussian processes on finite intervals are obtained recently in [61]; see also [62]. Our result above is of interest when considering the growth of supremum of on large increasing intervals and is proportional to for some (recall (2.10) which guarantees the positivity of ).
Wills functionals are tightly related with Borelltype bounds for tail distribution of suprema of Gaussian processes. We derive below a variant of Borell inequality accommodated to the setup of this section, see also [59][Cor 1]. For a given variance function , let in the following .
Proposition 3.5.
Suppose that is a centered separable Gaussian process with bounded sample paths and variance . For any such that and with we have
(3.9) 
for any .
Next, inserting in (3.9) we get a lower bound for Wills functional (3.5), namely
In the special case of having stationary increments, utilizing properties of Pickands constants, we arrive at the following corollary to Proposition 3.5. Set below , , with a standard fractional Brownian motion with Hurst index .
Corollary 3.6.
Suppose that is a centered Gaussian process with stationary increments, bounded sample paths and variance function . If for , with and , then for each
Note that is finite and asymptotically linear in as , by (3.7).
To this end we briefly discuss Piterbarg constants, which appear in the tail asymptotics of supremum of nonstationary Gaussian processes, see e.g., [63, 64].
Given as above, the corresponding Piterbarg constants are defined by
(3.10) 
for some measurable locally bounded function and or . So far Piterbarg constants with have been reported and their finiteness for general under some restrictions on is recently shown in [65][Lem 4.5] and [66][Prop 3.1]. Clearly, the main question that arises is if is finite. For , its finiteness is shown using Piterbarg’s approach, see [63]. We establish below the finiteness of Piterbarg constants for some general by using the fact that is BrownResnick stationary, and moreover derive a bound in terms of Wills functional. In the following proposition we consider only the case ; scenario follows by analogous line of reasoning.
Proposition 3.7.
If is as in Proposition 3.6, then for any measurable function such that and , we have that and moreover
(3.11) 
3.3. Dissipative and conservative stationary maxstable processes
The decomposition of maxstable processes into conservative and dissipative parts has been recently discussed in [6] under some assumptions on the spectral process ; see also the recent review [67] for processes. Previous results based on the works of Rosiński [68, 69] have investigated decompositions of sumstable and maxstable processes; see also [43, 34, 32]. Since we consider in this contribution the de Haan representation (1.1), we shall rely on the recent findings of [6]. In [45][Lem 16] it is shown that a conservative/dissipative decomposition of corresponds to the socalled cone decomposition. In view of [6] a simple criteria for a stationary maxstable process with locally bounded sample paths to be generated by a nonsingular conservative flow (abbreviate this as ” is conservative”) is , where is some spectral process of . The next two results propose conditions for as above to be conservative, or for to have a dissipative Rosiński representation. For simplicity, we consider the case that has càdlàg sample paths.
Proposition 3.8.
Let be stationary and maxstable process with Fréchet marginals and càdlàg sample paths. Denote by its spectral tail process. Then the following are equivalent:

is conservative.

The Pickands constant equals 0.


We state next equivalent conditions for to have a dissipative Rosiński representation.
Proposition 3.9.
Remark 3.10.
The above propositions hold also for the discretetime setup , substituting the integral by the sum respectively. Additionally, in view of [22] and [19] it follows that
is equivalent with to have a dissipative Rosiński representation, which is also equivalent with the anticlustering condition for introduced in [21], see also [70].
3.4. Rosiński and De Haan Representations
In view of [6] and the above discussions, simple conditions on or guarantee the dissipative Rosiński representation (2.3). When has such a representation, using either [6] or our results here, we can construct the random shape function explicitly. Then, applying for instance [46][Thm 4.2] we can determine spectral processes as in (2.4) by choosing some random variable with positive density being independent of . Note that in the aforementioned reference with continuous sample paths are considered. The same holds under a more general assumption that is measurable.
Theorem 3.11.
If is a measurable nonnegative process with locally bounded sample paths such that for some , then the maxstable processes corresponding to the spectral processes determined for different independent of via (2.4) are stationary with Fréchet marginals, equal in distribution, and have dissipative Rosiński representation (2.3).
Remark 3.12.
The constant in the representation depends only on . Indeed, since we assume that has Fréchet distribution, by (1.2)
(3.12) 
To this end we mention an important class of maxstable stationary processes for which is deterministic.
3.5. Growth of supremum
Below let be a separable maxstable stationary process with locally bounded sample paths and spectral process such that (1.1) holds. For any and we have for any
where stands for the Euler Gamma function and we used (1.3). Consequently, we have
(3.13) 
A direct implication of (1.8), (1.9) and (3.13) is the following proposition.
Proposition 3.13.
If where or is a maxstable stationary process as above, then (3.13) holds and moreover is uniformly integrable for any .
3.6. Maxstable random fields
Maxstable random fields can be defined exactly as the maxstable processes by simply substituting the random process by a random field . The corresponding Pickands constant is then defined by
where again we suppose that is stationary maxstable with Fréchet marginals . Since the functional is translation invariant, i.e.,
and is subadditive in the sense that for disjoint , then by [71]
we have that
exists and is finite.
In view of [6], with locally bounded sample paths has a dissipative Rosiński
representation if , where with the Lebesgue measure in . Our findings above can be easily extended to this setup of stationary maxstable random fields. Note in passing that for the discretetime setup the extremal index of is calculated in terms of spectral functions in [20], see also [31] for formulas in terms of spectral tail process.
Given the importance of the Gaussian model, for simplicity we state next the result for the BrownResnick maxstable model.
Let be a lognormal Gaussian random field where is a centered jointly measurable and separable Gaussian random field with stationary increments and variance function . If , which in light of [4] follows if
(3.14) 
where stands for the Euclidean norm, then we have
(3.15) 
Next, we discuss an important issue relevant for the simulation of BrownResnick maxstable stationary random fields, a topic of great interest for various applications, see for recent developments [72, 73, 9, 49, 74, 75]. As formulated in [75][Problem 2] for simulation of the BrownResnick maxstable stationary random field with spectral random field with a compact subset of , it is important to be able to minimise the functional
for all centered Gaussian random fields with stationary
increments and variance function that have the same variogram for some given continuous function .
We conclude this section with the following comparison result:
Proposition 3.15.
For two given separable centered Gaussian random fields with stationary increments, bounded sample paths and variance functions respectively, with a compact subset of , and for any measurable and locally bounded function , we have
(3.16) 
provided that .
4. Proofs
Proof of Theorem 2.1 We adapt the arguments of the proof of [2][Thm 2.1] for our maxstable process. Note first that by (1.11), in view of [6] has a dissipative Rosiński representation with some process which by the construction in the aforementioned paper is stochastically continuous and locally bounded (these properties are inherited from ). Therefore a jointly measurable and separable version of which is locally bounded exists, and we shall consider this version below.
Step 1: Since is given by (2.4), and moreover is locally bounded. then by (1.4) for any we have that
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